We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state ρ=v|ψ−⟩⟨ψ−|+(1−v)14 via a local hidden variable (LHV) model, where |ψ−⟩ denotes the singlet state. We show analytically that these correlations are local for v=999×689×10−6 cos2(π/50)≃0.6829 . In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constant KG(3)≤1/v≃1.4644 . We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state for v≃0.4553 . The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.
Grothendieck constants
KG(d) bound the advantage of
d-dimensional strategies over
1-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from
d=2, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For
d∈{3,4,5}, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for
d⩽9. For
d∈{4,7,8}, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants
KG(d↦2) as the advantage of complex
d-dimensional quantum mechanics over real qubit quantum mechanics. Motivated by this connection, we also improve the bounds on
KG(d↦2).
Abstract Grothendieck’s bound is used in the context of a single quantum system, in contrast to previous work which used it for multipartite entangled systems and the violation of Bell-like inequalities. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form C that uses complex numbers in the unit disc, and takes values less than 1. It then proves that if the complex numbers are replaced with vectors in the unit ball of the Hilbert space, then the ‘quantum’ quadratic form Q might take values greater than 1, up to the complex Grothendieck constant k G . The Grothendieck theorem is reformulated here in terms of arbitrary matrices (which are multiplied with appropriate normalisation prefactors), so that it is directly applicable to quantum quantities. The emphasis in the paper is in the ‘Grothendieck region’ ( 1 , k G ) , which is a classically forbidden region in the sense that C cannot take values in it. Necessary (but not sufficient) conditions for Q taking values in the Grothendieck region are given. Two examples that involve physical quantities in systems with six and 12-dimensional Hilbert space, are shown to lead to Q in the Grothendieck region ( 1 , k G ) . They involve projectors of the overlaps of novel generalised coherent states that resolve the identity and have a discrete isotropy.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
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